Then \(\Vert \cdot \Vert _{H^p}\)  is a norm for \(p\in [1,\infty )\) . It is easy to verify also that this is a \(p\) -norm for \(p\in (0,1)\) , in particular for \(\Vert \cdot \Vert _{H^p}^p\)  the triangle inequality holds. Every function \(f(z)\in H^p(\mathbb {D})\)  has boundary values \(f(e^{i\theta })\in L^p(\partial \mathbb {D})\)  and
the map \(f(z)\mapsto f(e^{i\theta })\)  is an isometrical isomorphism of \(H^p(\mathbb {D})\)  onto some closed subspace of \(L^p(\partial \mathbb {D},\frac{d\theta }{2\pi })\)  (see, e. g., [1]}, [2]}).
